2000 messages per day is a reasonable enough volume to get you blocked from the vast majority of Shared Web Hosts mail platforms, virtually guaranteed. They won't like you putting their Shared IP ranges at risk of being blocked. I can assure you that GMail, Yahoo, Hotmail et al blocking a Shared IP range is a massive pain in the behind - so I'm not hugely surprised that Gmail throw a fit too ... despite their size they'll suffer the same pain if other major ISP's start blocking their IP ranges.
What you need to do is look at third party solutions, there are a wealth of mass mail providers - most of them providing superb analytical detail on your mailshots, aswell as tools to help you evaluate how 'spammy' your message looks. You pay by the mail in most cases, so you should be able to find a solution which fits your budget.
A couple of examples which I have experience of, and can confirm are excellent platforms to work with are:
www.mailingmanager.co.uk - they offer a decent free trial to get used to it
www.streamsend.com - not used personally but heard good things from clients.
Also, this is probably stating the obvious but be very careful of who you are sending to. It doesn't matter how much money you throw at these companies, they will not tolerate spammers - make sure your lists are clean, legitimate and opt-in only. Buying in lists is tantamount to trying to smuggle the Devil into Heaven with most of these companies.